On Page 64, Set Theory, Jech(2006), define the following, by transfinite induction:

$V_0=\emptyset$,

$V_{\alpha+1}=P(V_{\alpha})$,

$V_{\alpha}=\bigcup_{\beta<\alpha}V_\beta$, if $\alpha$ is a limit ordinal.

How can we prove $V_{\alpha}$ is transitive by induction.

I tried what if other set, say $\{\{\emptyset\}\}$, is disignated as $V_0$. It turns out the transitive property fails. So somehow I should incoorperate $V_0=\emptyset$ into the induction. But then I stared at it, I stared at it, I just don't know what should I do.

If $V_{\alpha}$ is transitive, consider $x \in y \in P(V_{\alpha})$. Then $y$ is a subset of $V_\alpha$, so that $x$ is in a subset of $V_{\alpha}$ and hence is an element of $V_{\alpha}$, so must also be an element of $P(V_{\alpha})$. Thus $P(V_{\alpha}) = V_{\alpha+1}$ is transitive.